Chapter 9.2, Problem 88AYU

Explain why the vertical-line test used to identify functions in rectangular coordinates does not work for equations expressed in polar coordinates.

To determine

Write the two different tests for symmetry with respect to the polar axis. Give examples in such a way that one test works and the other fails. Conclude the most preferable test with justification.

Answer to Problem 88AYU

$- r = \text{sin}\left(\pi - 2\theta \right) = \text{sin}\pi \text{cos}2\theta - \text{cos}\pi \text{sin}2\theta = \text{sin}\left(2\theta \right)$

Explanation of Solution

Given:

It is asked to write the two different tests for symmetry with respect to the polar axis. Give examples in such a way that one test works and the other fails. Conclude the most preferable test with justification.

Check for symmetry test:

Polar axis:

If the point $\left(r, \theta \right)$ lies on the graph, then the point $\left(r, - \theta \right)$ or $\left(- r, \pi - \theta \right)$ also lies on the graph.3

Consider the example $r = \text{sin}\left(2\theta \right)$,

Replace $\theta$ by $- \theta$,

$r = \text{sin}\left(2\left(- \theta \right)\right) = - \text{sin}\left(2\theta \right).$

This test fails.

Replace $r$ by $- r$ and $\theta$ by $\pi - \theta$,

$- r = \text{sin}\left(\pi - 2\theta \right) = \text{sin}\pi \text{cos}2\theta - \text{cos}\pi \text{sin}2\theta = \text{sin}\left(2\theta \right)$

Consider the example $r = \text{sin}\left(2\theta \right),$

Replace $\theta$ by $- \theta$,

$r = \text{sin}\left(2\left(- \theta \right)\right) = - \text{sin}\left(2\theta \right).$

This test fails.

Replace $r$ by $- r$ and $\theta$ by $\pi - \theta ,$

$- r = \text{sin}\left(\pi - 2\theta \right) = \text{sin}\pi \text{cos}2\theta - \text{cos}\pi \text{sin}2\theta = \text{sin}\left(2\theta \right)$